The bilateral symmetry
of the human face is known to be a considerable factor in its attraction.
In fact, "averaging" the two sides to induce even greater symmetry increases
the attraction (Langlois & Roggman, 1990). Even
though the two sides of the face are not perfectly symmetric, the near
symmetry is undeniably attractive. To attempt to assess the degree of bilateral
symmetry in the human face one can separate the face into its two halves,
left L and right R, then flip R in a horizontal reflection
to make the halves directly comparable, and finally take the symmetric
difference of these two sets for comparison. The situation is rather like
that depicted in Figure 7.042 of The Fascination of Groups (Budden,
1972: 64), where the symmetric difference of two sets A and B,
which are approximately the same, is depicted. When the two reach identity,
the symmetric difference ("one or the other but not both") becomes the
null set Ø. On the other hand, the greater the symmetric difference,
the greater the degree of asymmetry.

Now it also happens that
the symmetric difference is one of the two universals for the category
of sets (Kostrikin & Shafarevich, 1990: Sec. 20),
the other universal being the Cartesian product ´
. As indicated above, it is of both a symmetric and an asymmetric nature
in that it measures the degree of equality of two sets, one of which may
be a transform of the other. The symmetric difference also plays a key
role in the homology of sets as the boundary operator ¶
.

Given the aesthetic satisfaction
that accrues to facial symmetry and to symmetry in general, one is led
to wonder whether the symmetric difference may perhaps be involved in a
significant way in the general psychology of perception. In a sense this
is close to Jablan’s (2002) modularity in science and
art, Avital’s "mindprints" (Avital, 1999), and to Markovich’s
amodal completion of visual perception (Markovich,
2002), wherein the antinomy, symmetry-asymmetry, plays an important role.
The symmetric difference, by its very nature, measures symmetry versus
asymmetry. However, with respect to aesthetic satisfaction, the situation
seems to depend more on Riegel’s dialectical psychology (Riegel, 1973,
1977,
1978).
Dialectical psychology posits that we go through life continually encountering
and reconciling contradictions, which arise out of the classical Heraclitean
dialectic of change and incompleteness. As Riegel put it, "Thinking is
the process of transforming contradictory experience into momentary stable
structures."

The symmetric difference and the
dialectical pair

Dialectic is a word of
many meanings (Rychlak, 1976) but taken here as number
(8) in the article "Dialectic" in the Encyclopedia of Philosophy (Edwards,
1967), namely, as "…the logical development of thought or reality through
thesis and antithesis to a synthesis of these opposites." The essence of
dialectic thus lies in the apposition of opposites—thesis and antithesis—that
through removal of sensed contradictions arrives at a synthesis, at a higher
level perhaps. Associated with every figure, there is a ground. Doubt surrounds
every belief: "It could be that way, but is it really?" To every appearance,
there is the reality—"the thing in itself." Emotion stems from a primitive
approach-avoid reaction. Intelligence involves an exploration of alternatives.

Dialectical thinking
consists of an exploration of contradictory possibilities that results
in a cognition that reduces cognitive dissonance. Kahneman
and Miller (1986) state that all perceived events are compared to counterfactual
alternatives, counterfactual in that they constitute alternative realities
to the one experienced. Thus even without the element of ceaseless change—Heraclitus’s
"everything flows"—the distinction between an object or concept and what
it is not leads inevitably to a dialectic view of the world.

Following Hegel’s axiomatization
of dialectic, Riegel (1973) laid down the following
three "laws" of dialectical psychology:

I. The unity and struggle of opposites
II. The transformation of quantitative into qualitative change
III. The negation of the negation

Here these three "laws"
will be expressed, in the contexts of both dialectic and dialectical psychology,
by the symmetric difference and its complement. Formally the symmetric
difference of two sets A and B is defined by

ADB
= (AÇ
¬ B) È
(¬ AÇB)
= (AÈB)
\ (AÇ B)
= ¬ AD
¬ B, (1)

while its complement,
symbolized by ¬, is

¬ (ADB)
= (AÇB)
È
¬ (AÈB)
= (AÇB)
È
(¬ AÇ
¬ B) = ¬ ADB
= AD
¬ B. (2)

The pair of relations
(1) and (2) will be referred to as the dialectical pair (D,
¬). The relation to dialectic is made plain below.

The symmetric difference
operation generates a group upon the category of sets. The null set Ø
is the identity: AD
Ø = A. It is self-inverse: ADA
= Ø, and associative and commutative: AD
(BDC)
= (ADB)
DC,
and ADB
= BDA.
The property

ADW
= ¬ A, where W
denotes the universe of discourse,

suggests that complementation
of the symmetric difference to form the dialectical pair is a natural operation.

Clearly, Eq. (1) for
the symmetric difference—"one or the other, but not both"— embodies "law"
I. Since both Eqs. (1) and (2) are set-theoretic, they are in accord with
"law" II, transformation from quantities to qualitative sets. (But according
to a theorem of Nikodym (1930), there also exists a
means for going back from the sets themselves via a measure on the sets
determined by the symmetric difference. Any world view that would lack
measures for geometry and probability must be seriously defective.)

Ordinarily "law" III
is taken to constitute simple negation. A truly new aspect of "law" III,
however, is provided by the dialectical pair formulation in that it is
not simple negation but rather the complement of the symmetric difference
(1), as expressed by Eq. (2), that corresponds to "law" III. This not only
permits synthesis in the product term but also relates the sets involved
to everything else in the domain of discourse—their context.

Consider then the dialectical
properties of the dialectical pair. Hegel’s first triad was "being—nothing—becoming."
" Being" is positive; it affirms existence of an object or concept
C,
which is always accompanied by its complement in the universe of discourse
W.
The symmetric difference of an object with its complement "becomes"—is
everything, namely, the universe of discourse.

CD
¬ C = (CÇ
¬ ¬ C) È
(¬ CÇ
¬ C) = (CÇC)
È
(¬ CÇ
¬ C) = CÈ
¬ C = W.

The second element of
the triad, "nothing," is negative; it denies what is being affirmed and
is deduced directly from "being," as the null set Ø. This is given
by the symmetric difference of an object with itself,

CDC
= (CÇ
¬ C) È
(¬ CÇC)
= Ø È
Ø = Ø.

Passage of "nothing"
into "being" is generated by the symmetric difference:

(Ø DC)
= (Ø Ç
¬ C) È
(¬ Ø ÇC)
= Ø È
(WÇC)
= C

If C should happen
to be contained within a larger set C', CÌC',
then
C ÈC'
is the relative complement of C in C'. Thus D
acts as differentia in Hegelian dialectic.

The counterpart in formal
logic of the set relation (1) is (pÙ
~ q) Ú
(~ pÙq).
This expression differs from the Sheffer stroke only by lacking a join
with the joint denial term ~ pÙ
~ q. Since all of first order logic can be derived from the Sheffer
stroke, this relation between logic and this set-theoretic universal is
worthy of note. The second expression (2) corresponds to the first order
logic expression for equivalence, (pÙq)
Ú
( ~ pÙ
~q), thus providing a basis for qualia. The dialectical pair thus
provides a basis for logical thought as well as the more natural intuition
involved in dialectical psychology and psychological categorization.

Before proceeding to
the role of the dialectical pair in dialectical psychology, we note parenthetically
that Hegel himself would no doubt have found such formal modeling of dialectic
as that above an abomination. Styazhkin (1969: 112)
comments in connection with Hegel’s denunciation of Ploucquet’s "logical
calculus" that "One can only imagine what epithets Hegel would have
bestowed on contemporary mathematical logic!" Yet Ploucquet’s calculus
for generating a complete set of all logical relations out of only an identity
function and an inconsistency function seems rather close to Hegel’s
own ideas on thesis-antithesis and synthesis.

The Dialectical Pair and Dialectical
Psychology

The "similarities and
differences" paradigm is an old story in psychology. Clearly classification
depends upon detecting differences, which may then be used to separate
objects and concepts into categories of exemplars determined by their closeness
to the prototypes. Survival is more dependent upon recognizing novel elements
of the environment and classifying them than contemplating their meaning
and significance. The commonality of two such stimuli, C Ç
C', corresponds to convergent thinking. "Everything else," embodied in
the ¬ (CÈC'
) term in Eq. (2), represents divergent
thinking. The dialectical pair, Eqs. (1) and (2), thus encompasses both
the "similarities-differences" paradigm and convergent and divergent thinking.
As Riegel (1973: 351) put it,

and then, by connecting
them in a systematic manner, tries to reconstruct

the phenomena … in
an unequivocal manner… . Dialectical thinking

comprehends itself,
the world, and each concrete object in its multitude of

of contradictory relations.

Let us now flesh out
the process described in the opening paragraph for determining the degree
of bilateral symmetry in a human face. Denote by R' the result of
flipping the right half R of the face horizontally. Then the symmetric
difference LDR'
describes
by its nearness to the null set Ø the degree of closeness in appearance.
However, the second half of the dialectical pair brings in more aspects
of cognitive processing. The complement ¬ (LDR
')
= (LÇR
'
) È
¬ (LÇR
')
not only describes this closeness by the commonality expressed in the first
term on the right but also refers the result to the context—not-(L
and/or R' )—in the current
universe of discourse. The latter will ordinarily include more than shape,
bringing in memory, both cognitive and emotional and mental sets that predispose
to certain features that are found attractive. Proverbially, "The eyes
are the windows of the soul." Such facial features as this that occur within
the symmetric pattern also enter into attractiveness. Furthermore, "beauty
is in the eye of the beholder." The particular cognitions and emotions
within one’s psyche which impart personal aesthetic satisfaction thus factor
in also. I have shown elsewhere (Hoffman, 1995, 1997,
1999)
that the cognitive factors of categorization, memory (both long-term and
working), language, decisionmaking, learning, problem solving, intelligence,
and creativity are structured by the dialectical pair. Both psychological
categories and mathematical categories stem from equivalence (Mervis
& Rosch, 1981; Eilenberg & MacLane, 1945)
if emphasis is shifted in the case of psychological categories to functions
and away from objects and their attributes. Psychological categories consist
of objects with particular attributes structured by prototypes and exemplars
that are distinguished from near-neighbor categories by gradients. The
system strongly resembles the idea of suspension in topology (Hatcher,
2002). A suspension consists of a double cone joined at the base, with
the vertices consisting of the two different prototypes. A mathematical
category consists of a collection of objects together with, for each pair
thereof, a collection of morphisms, which are arrows mapping objects to
one another in an associative way. Mathematical categories also have their
objects but the emphasis is placed upon the morphisms which generate the
structural diagrams of the category, as in the following commutative diagram
in Figure 1for the dialectical pair:

"Chasing around the diagram"
is fundamental in category theory and algebraic topology. It is apparently
reflected in the mind’s "trains of thought" as chasing around the cognitive
diagrams involved in "brain circuits."

Memory represents the
store of information derived from experience. It is described as sensory,
short-term, long-term, working, declarative versus procedural, semantic
versus episodic, explicit (conscious) versus implicit (subconscious), etc.
Semantic memory is modeled in terms of networks whose vertices are "chunks"
and arrows are directed "schema," and so have the geometric nature of a
simplicial triangulation of a space or a simplicial complex. A simplicial
complex consists of a finite set of simplices which form linear combinations
called chains, the boundaries of which are generated by the action of the
boundary operator ¶
acting upon the chains. The standard models for memory processing (Chang,
1986) consist of networks which imply that long-term memory L has
the character of an ordered simplicial complex. As such, L must
contain subcomplexes Lj which correspond to particular
memories, semantic or episodic, in L. The set of all such subcomplexes
constitutes the power set of L. Suppose v denotes a memory
vertex-chunk in some Lj. The star of v, St(v),
is defined as the interiors of all those simplices that contain v.
The connection with the dialectical pair is immediate:

[Lj DSt(v),
¬ (LjDSt(v))]
= [Lj\ St(v), St(v)
È
(L \ Lj)].

The first term is the
complement in Lj . If it is the null set, recall is perfect.
Otherwise the second factor acts to have St(v) scan within
long-term memory outside of Lj, i.e., to search memory
further.

Conscious working memory
W
consists of the conscious boundary of long-term memory L in combination
with short-term memory S: W = ¶LÇS.
It interacts with the concepts in the working memory from the previous
moment and the percepts forwarded via short-term memory from the sensory
buffer. Working memory is path-connected. Each path corresponds to a different
constituent of the working memory field, thus offering a basis for the
decision making among alternatives that constitutes the cognitive executive
function. This continuous flow from one working memory to the next thus
provides the "flashing stream" (Dodwell, 2001) of consciousness.

Intelligence refers to
the mental ability to reason, comprehend complicated concepts, think abstractly,
"transfer" (that is, generalize from experience), and arrive at sound decisions.
Whatever intelligence may connote, it is intimately related to working
memory W (Eysenck, 1967, Wickelgren,
1997). The close connection has also been established by neuroimaging (Gabrieli,
1998). She comments, "Reasoning seems to be the sum of working-memory abilities".
Intelligence acts to strip away confusion, to be "a quick study," and arrive
at the heart of the matter within the context at hand. This of course is
of the nature of the dialectical pair, whose first component strips away
the overlap of concepts and whose second component penetrates quickly to
the commonality and exploration of alternatives that characterizes intelligence.
The first phase of the dialectical pair, Eq. (1), acts to diminish any
confusing common attributes. All that remains after the application CDC'
is the separate character of each with no overlap—no "confusion." Here
the cognitive dissonance generated by the first stage is resolved by whatever
C
and C' may have in common in company with their context in memory
and working memory.

Emotion And Dialectic

Presented with an ostensibly
aesthetic object, one reacts with either like or dislike, for all the world
like the approach-avoidance reaction fundamental to emotion. Human emotion
has evolved over long eons out of the basic approach-avoidance reaction
found at the level of the reptilian brain. This "survival brain" within
the primitive midbrain has added higher centers during the course of evolution:
the limbic system, or "emotional brain," and the coldly calculating neocortex,
which acts as a brake upon basic reactions. Emotions separate into two
major groups, the unpleasant ones of anxiety, disgust, hate, and anger
in one, and the pleasant emotions—joy, pride, hope, and affection—in the
other.

This bipolar nature of
emotion lends itself nicely to the dialectical pair. Emotion E distinguishes
itself from emotion E' via EDE
',
while at the same time ¬ (EDE
'
) permits one to love and hate at the same time. The bipolar scale between
approach and avoid is termed emotional valence. The accompanying emotional
intensity scale is called the activation level. The stronger the activation
and valence, the more an emotion is likely to be felt as a "pure" emotion.

Emotion emerges as a
more less surprising change from a core of calmness that corresponds to
Campos
and Caplovitz’s (1988) normative "invariant core of affective continuity."
As Ellsworth (1994) puts it, "The possibility of a
valenced response—‘this event may be good or bad for me’—is enough to elicit
emotion." The situation is depicted in the symmetric tetrahedral patterns
of Figure 2.

Calm Awareness

Figure 2. Paired tetrahedral representation
of the generation of bipolar emotions by surprise out of a "core of calmness."

An aesthetically satisfying
object is not just an object. It has form and meaning. We have dealt with
the cognitive aspect above. Here we examine the psychological basis for
aesthetic form and claim that the well known relation in physical science:

invariance Þ
symmetry
Þ
conservation law

carries over here to
where the invariances are those of the psychological constancies. Psychological
conservation resides in the continuous symmetry of the constancies and
form memory. Psychological constancy in the visual case refers to the invariant
recognition of perceived objects no matter what distortion may be imposed
on them by viewing conditions: location in the field of view, rotation,
binocular perception, apparent size, state of motion (as long as they are
not "moving too fast for the eye to follow"), and color. Pitch and loudness
constancies and binaural perception constitute similar invariances for
audition. The visual constancies, their associated continuous transformation
groups, Lie derivatives, and local orbit structures are listed in Table
I (Hoffman, 1966a, 1978, 1984).
Recall that invariance under a continuous (Lie) transformation group means
annulment of the invariant by the Lie derivative and that the local orbit
structure is determined by means of the associated Pfaffian system.

Good design is thought
to reside not so much in the realization of form but in the elimination
of "misfit" (Alexander, 1967). That is, there is a
subconscious response to forms that are found naturally in the perceptual
system, like the constancy orbit patterns. "Love at first sight" involves
immediate strong recognition of some subconscious ideal pattern. Certain
basic structures in the neuropsychology of form perception are known to
embody the patterns of the psychological constancies (Van
Essen and Gallant, 1994). Ornamentation, primitive as well as modern,
exhibits many of these basic patterns. Optical Art also involves these
basic patterns (Hoffman, 1966b), and the Golden
Ratio postulated to govern the aesthetic properties of facial and body
dimensions ( www.goldennumber.net) also makes its appearance as the logarithmic
spiral generated by the combined action of size constancy (the dilation
group in log r ) and the rotation component of shape constancy,
with rotation angle j
.

.
(3)

According to (Pronzato,
Wynn, and Zhigljavskii, 2000). this Golden-Ratio combination serves
for optimal search of the visual field. See Huntley
(1990) for a full discussion of the wide role of the equiangular spiral
in nature and beauty.

I suggest that this harmony
among the Golden Ratio, the aesthetically satisfying spirals that are found
throughout nature, and the psychological constancies constitutes the psychological
basis for the pleasurable perceptions imparted by the Golden Ratio. Gary
Meisner’s website www.beautyanalysis.com
shows a collection of images depicting the role of the Golden Ratio in
nature, the human body, and the human face and supports a strong claim
that Phi , the Golden Ratio, is the key to beauty. Michael Semprevivo
(http://goldennumber.net/color.htm)
has developed a software called PhiBar to express in terms of the
Golden Ratio the color spectrum relationships that produce rich, appealing
color combinations.

But when one considers
the stages that art has gone through from shamanic cave art thousands of
years ago to abstract expressionism of today, it is clear that fashion
and learning impose considerable influences upon such intrinsic psychological
systems as the constancies and Gestalt "laws." There is often a conflict
between the art appreciation of one generation and the next. When "les
Fauves" first burst upon the scene in Paris in 1905, the art critics and
the public were decidedly negative. Yet this furor over non-traditional
use of contour and color gradually subsided and in time was replaced by
appreciation for the impressionist school (though an educated taste might
still be required). Cognitive and emotional influences are as much involved
in art appreciation as perceptual ones, perhaps even more so. The emotional
impact of an art object or style may well be negative at first sight but
after enough learned meanings have been assimilated and associated with
the structural elements consonant with the basic perceptual symmetries
an appreciation of the art form develops, if indeed the latter possesses
any aesthetic import at all. Given the progressive interactions during
this learning process among emotion E, long-term memory L,
and aesthetic satisfaction A, the triple symmetric difference relation
(EDL)
Ç
(LDA)
Ç
(ADE)
= Ø (see Appendix) represents by its annulment the commonality of
acceptance.

But any theory that purports
to offer a psychological basis for art appreciation must account for such
abstract expressionist paintings as those of Jackson Pollock. These action
paintings, uncharitably attributed to "Jack the dripper," consist of lines,
curves, and colors that derive from the painter’s wrist motions and dancing
about while flinging paint. The elements of the human psyche that respond
to such formless patterns must be very basic indeed, even more so than
for cave art and Paleolithic ornamental art. If so, it must involve invariances
and so symmetries that are very fundamental indeed to the human condition,
something like Gibson’s "formless invariants." The writer (Hoffman, 2001)
has suggested that the fundamental geometric element of curvature is one
such, and here I offer the suggestion that Heraclitus’s panta rhei—everything
flows—is another. A flow represents the local orbit structure of a Lie
transformation group, perceived here as the vigorous flow of color and
form responding to psychological constancy independent of cognitive recognition.
The presence of the constancies in the human perceptual system is what
accounts for abstract expressionism.

Appendix: Table of Symmetric Difference
Relations

E X P R E S S I O N (Daniell,
1917)

The symmetric
difference operation D
generates a group on sets {A}, the identity being the null set Ø.
The group is self-inverse: ADA
= Ø. The operation is associative

AD
(BDC)
= (ADB)
DC,
and commutative ADB
= BDA.

ADB
= (AÈB)
\ (AÇB)
= (AÇ
¬ B) È
(¬ AÇB)
= ¬ AD
¬ B.

¬ (AD B)
= (AÇB)
È
¬ (AÈB)
= (AÇB)
È
(¬ AÇ
¬ B) = ¬ ADB
= AD
¬ B.

ADB
= Ø if and only if A = B.

If ADB
= C, then A = BDC
and B = ADC.
Note: This expression provides a means for solving for an unknown set.

Campos, J. J. and B. Caplovitz, A
new understanding of emotions and their development. In Emotions,
Cognition, and Behavior, Izard, C. E., J. Kagan, and R. B. Zajonc (Eds.),
Cambridge University Press, New York (1988).

Dodwell, P. C. Brave New Mind,
Oxford University Press, New York (2001).

Edwards, E. (Ed.), Encyclopedia of
Philosophy, Macmillan, New York (1967).

Eilenberg, S. and MacLane, S. General
theory of natural equivalences, Transactions of the American Mathematical
Society 58 (1945) 231-294.

Ellsworth, P. C., Some reasons to
expect universal antecedents of emotions. In The Nature of Emotion:
Fundamental Questions, Ekman, P. and R. J. Davidson (Eds.). Oxford
University Press, New York (1994).